ELASTICITYDemonstrate your understanding of elasticity, elastic limit, stress, strain, and ultimate strength.Write and apply formulas for calculating Youngs modulus, shear modulus, and bulk modulus.

Elastic Properties of MatterAn elastic body is one that returns to its original shape after a deformation.

Elastic Properties of MatterAn inelastic body is one that does not return to its original shape after a deformation.

Elastic or Inelastic?An elastic collision loses no energy. The deform-ation on collision is fully restored. In an inelastic collision, energy is lost and the deformation may be permanent. (Click it.)

Hookes LawWhen a spring is stretched, there is a restoring force that is proportional to the displacement.F = -kxThe spring constant k is a property of the spring given by:The spring constant k is a measure of the elasticity of the spring.

Types of StressA tensile stress occurs when equal and opposite forces are directed away from each other.A compressive stress occurs when equal and opposite forces are directed toward each other.

Summary of DefinitionsStress is the ratio of an applied force F to the area A over which it acts:Strain is the relative change in the dimensions or shape of a body as the result of an applied stress:Examples: Change in length per unit length; change in volume per unit volume.

Longitudinal Stress and StrainFor wires, rods, and bars, there is a longitudinal stress F/A that produces a change in length per unit length. In such cases:

The Elastic LimitThe elastic limit is the maximum stress a body can experience without becoming permanently deformed.If the stress exceeds the elastic limit, the final length will be longer than the original 2 m.OkayBeyond limit

The Ultimate StrengthThe ultimate strength is the greatest stress a body can experience without breaking or rupturing.If the stress exceeds the ultimate strength, the string breaks!

The Modulus of ElasticityProvided that the elastic limit is not exceeded, an elastic deformation (strain) is directly proportional to the magnitude of the applied force per unit area (stress).

Youngs ModulusFor materials whose length is much greater than the width or thickness, we are concerned with the longitudinal modulus of elasticity, or Youngs Modulus (Y).

Example 1: Youngs modulus for brass is 8.96 x 1011Pa. A 120-N weight is attached to an 8-m length of brass wire; find the increase in length. The diameter is 1.5 mm.First find area of wire:A = 1.77 x 10-6 m2

Example 2: (Continued)Y = 8.96 x 1011 Pa; F = 120 N; L = 8 m; A = 1.77 x 10-6 m2 F = 120 N; DL = ?DL = 0.605 mmIncrease in length:

Shear ModulusA shearing stress alters only the shape of the body, leaving the volume unchanged. For example, consider equal and opposite shearing forces F acting on the cube below:The shearing force F produces a shearing angle f. The angle f is the strain and the stress is given by F/A as before.

Calculating Shear ModulusThe strain is the angle expressed in radians:Stress is force per unit area:The shear modulus S is defined as the ratio of the shearing stress F/A to the shearing strain f:The shear modulus: Units are in Pascals.

Volume ElasticityNot all deformations are linear. Sometimes an applied stress F/A results in a decrease of volume. In such cases, there is a bulk modulus B of elasticity.The bulk modulus is negative because of decrease in V.

The Bulk ModulusSince F/A is generally pressure P, we may write:Units remain in Pascals (Pa) since the strain is unitless.

Summary: Elastic and Inelastic

Summary of Definitions

Longitudinal Stress and Strain

Youngs Modulus

The Shear Modulus

The Bulk Modulus

P-Wave(Body Wave) Primary or compressional (P) waves a) The first kind of body wave is the P wave or primary wave. This is the fastest kind of seismic wave. b) The P wave can move through solid rock and fluids, like water or the liquid layers of the earth. c) It pushes and pulls the rock it moves through just like sound waves push and pull the air.d) Highest velocity (6 km/sec in the crust)

P-Wave

Secondary Wave (S Wave) Secondary or shear (S) waves a)The second type of body wave is the S wave or secondary wave, which is the second wave you feel in an earthquake.b) An S wave is slower than a P wave and can only move through solid rock. (3.6 km/sec in the crust) c) This wave moves rock up and down, or side-to-side.

S-Wave

L-WaveLove WavesThe first kind of surface wave is called a Love wave, named after A.E.H. Love, a British mathematician who worked out the mathematical model for this kind of wave in 1911. It's the fastest surface wave and moves the ground from side-to-side.

L-Wave

Rayleigh WavesRayleigh WavesThe other kind of surface wave is the Rayleigh wave, named for John William Strutt, Lord Rayleigh, who mathematically predicted the existence of this kind of wave in 1885. A Rayleigh wave rolls along the ground just like a wave rolls across a lake or an ocean. Because it rolls, it moves the ground up and down, and side-to-side in the same direction that the wave is moving. Most of the shaking felt from an earthquake is due to the Rayleigh wave, which can be much larger than the other waves.

Rayleigh Waves

Seismic Wave Speeds

Propagation of the seismic wave:Seismic waves subjected to several phenomenon when travel through the earthThe most important are:1- Attenuation2- Reflection3- Refraction4- Diffraction

1- AttenuationThere are two types of attenuation:

Geometrical spreading: Take place due to traveling certain amount of distanceExample: Find attenuation of a wave after a distance (r) from the sourceIo, roI= Io * ro/ r * e- r is absorption coefficient5- Multiples6- Generation of wave face7- Change of velocity8- Frequency filtering

2- Reflection:It is take place when a seismic wave hits an interface separating two media of different elastic Properties (or different acoustic impedance, z)

Acoustic impedance: define as the product of velocity with densityHigher frequencies attenuate over shorter distances due to their shorter wavelengths.Therefore, high frequencies decay first leaving a low frequency signal remaining.Note:Z = p * V

Reflection Coefficient: It is a ratio of reflected wave amplitude (Ar) to incidence wave amplitude (Ai)

R = Ar / Ai = Z2 Z1 / Z2 + Z1 = P2V2 P1V1 / P2V2 + P1V1

Notes:1- R is positive when Z2>Z1 and negative when Z1>Z22- R =+1 when Z1 = 0 and R = -1 when Z2 = 03- R is approach to unity in two cases: a- When incidence angle = Critical incidence angle b- Tangential (Grazing) incidence

3- Refraction : Apart of seismic wave is refracted when hits an interface separating two mediaRefraction depend on Snells law:

Notes:1- When V2 is smaller than V1 so i1> i2, in this case refraction will not take place, the wave will be deflected.

2- When V2>V1, i2 will be greater than i1 , when i2=90 the wave will travel along the interface and refraction will take place, So i2 is called critical angle.

4- Diffraction: it takes place when the seismic wave hits:

1- Irregularity2- Abrupt discontinuity3- FaultsIn this case the irregular feature act as point source for radiating waves in all directions.SourceSurfaceV1V2 Faults5- Multiple: They are signals undergone more than one reflection, and they are of small energy

There are two types of multiples:

1-Short path multiples : They are almost arrived with useful signals and form a tail to them, such asA- GhostB- Near-surface multiplesC- Peg-leg Multiples

6- Generation of wave phase:

When P-wave hits an interface generate four types of the seismic waves:Reflected P-WaveReflected S-WaveRefracted P-WaveRefracted S-WaveSeismic WaveNotes:1- When the wave hits an interface vertically , does not generate other type of waves.2- When the first medium is liquid, only three types will generate because S-wave does not propagates through the liquid.

Reflection: Basic GeometryT2 = X2 + (2Z/V)2 V2 T2 = X2 + T02 V2

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